Just off the top of my head, their 3rd string RB is a 5* from California. They convinced a 4* ESPN Top 100 DE to GREYSHIRT. He was from Louisiana, I believe. He stated in an interview "It's better to go to Alabama and sit on the bench than start at LSU." That seems dumb in a vacuum, but he elaborated that he knew he'd become a better player at Alabama, and that they play for championships, and no matter what LSU had going for it, they couldn't give him that. He didn't want our scholarship.
No, LSU cannot compete with that, when guys with offers to 20 other schools would essentially rather be a walk-on for a year.